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Given a vector bundle of rank , and any representation : (,) into a linear group (), there is an induced connection on the associated vector bundle =. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of E {\displaystyle E} and using the theory of principal bundles.
An Ehresmann connection on a fiber bundle (endowed with a structure group) sometimes gives rise to an Ehresmann connection on an associated bundle. For instance, a (linear) connection in a vector bundle E, thought of giving a parallelism of E as above, induces a connection on the associated bundle of frames PE of E.
A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.
The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP/G.
If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then define a connection that is compatible with this bundle metric, this is the metric connection. For the special case of E being the tangent bundle TM, the metric connection is called the Riemannian connection.
Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are sometimes called (vector) bundle homomorphisms. A bundle homomorphism from E 1 to E 2 with an inverse which is also a bundle homomorphism (from E 2 to E 1) is called a (vector) bundle isomorphism, and then E 1 and E 2 are said to be ...
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. [1] This is equivalent to: A connection for which the covariant derivatives of the metric on E ...
If is a vector bundle, there is one-to-one correspondence between linear connections on and the connections on the ()-module of sections of . Strictly speaking, ∇ {\displaystyle \nabla } corresponds to the covariant differential of a connection on E → X {\displaystyle E\to X} .